We report the results of a new measurement of spin structure functions of the deuteron in the region of moderate momentum transfer [Q2=0.27–1.3(GeV/c)2] and final hadronic state mass in the nucleon resonance region (W=1.08–2.0GeV). We scattered a 2.5 GeV polarized continuous electron beam at Jefferson Lab off a dynamically polarized cryogenic solid state target (15ND3) and detected the scattered electrons with the CEBAF large acceptance spectrometer. From our data, we extract the longitudinal double spin asymmetry A|| and the spin structure function gd1. Our data are generally in reasonable agreement with existing data from SLAC where they overlap, and they represent a substantial improvement in statistical precision. We compare our results with expectations for resonance asymmetries and extrapolated deep inelastic scaling results. Finally, we evaluate the first moment of the structure function gd1 and study its approach to both the deep inelastic limit at large Q2 and to the Gerasimov-Drell-Hearn sum rule at the real photon limit (Q2→0). We find that the first moment varies rapidly in the Q2 range of our experiment and crosses zero at Q2 between 0.5 and 0.8(GeV/c)2, indicating the importance of the Δ resonance at these momentum transfers.